A voyage through space(s) (Part III)

by | Oct 2, 2017 | Pure Maths | 0 comments

As promised, we begin our quest today in understanding what a normed vector space is. We will begin with a vector space V, and we would like to give it some structure similar to our metric spaces. Now we could of course give any metric to V, but we already have two operations of vector addition and scalar multiplication defined on our vector spaces, so we would really like our metric to be somewhat related to these. Let’s first deal with vector addition. We know that if we take two vectors x,y and translate them by the same amount a, the distance between them is the same. That is

    \[d(x+a,y+a)=d(x,y).\]

Now for scalar multiplication. We know that if we take our same two vectors x,y and the scale them by some scalar \lambda, that the distance between them also scales by \lambda. That is

    \[d(\lambda x,\lambda y)=|\lambda|d(x,y).\]

Now there is a special type of function that satisfies these two properties, namely a function called the norm. A norm  (\norm{.})is a function that takes elements of our vector space V to [0,\infty) such that the following three axioms are satisfied:

  1. \norm{x+y}\leq\norm{x}+\norm{y}
  2. \norm{\lambda x}=\lambda\norm{x}
  3. \norm{x}=0 \iff x=0

where x,y\in V and \lambda\in\mathbb{R}~ \text{or}~\mathbb{C}, depending on what your field your vector space is over. This norm is similar to the length of a vector in Euclidean space, the distance it is from the origin, however, since we can deal with a lot more general spaces, such sequence spaces and function spaces, the norm is more of a generalisation of this idea rather than an extension. In fact, in \mathbb{R}^n, where all the nice vectors that we like live in, the way we measure the length of these vectors is just a norm!

So what we have right now is a normed vector space V; a vector space with some norm defined on it. So for the rest of this post, I’m just going to go through some examples of normed vector spaces, just to get a grip on things. If you would like to prove that each of these actually define norms, try it out!

The absolute value function \abs{.} on \mathbb{R} is a norm, making \mathbb{R} a normed space. Look at that, there was right under your nose your entire life!

Now, what’s the next simplest type of space you could think of? Well, what about \mathbb{R}^N? We already know that these geometric vectors have the Euclidean norm defined as

    \[\norm{\vb{a}}_2=\big(\sum_{i=1}^{N}\abs{a_i}^2\big)^{\frac{1}{2}}.\]

However, what you have been calling the euclidean norm your life is also called the 2-norm, which is a specific case of the more general p-norm defined as

    \[\norm{\vb{a}}_p=\big(\sum_{i=1}^{N}\abs{a_i}^p\big)^{\frac{1}{p}},\]

where p\geq1. So as another example, here is the 1-norm: 

    \[\norm{\vb{a}}_1=\big(\sum_{i=1}^{N}\abs{a_i}\big).\]

The infinite norm is defined as

    \[\norm{\vb{a}}_{\infty}=max_{1\leq n \leq N}\abs{a_i}}.\]

The next type of normed vector space we would like to deal with are sequence spaces. Sequences can be added together and multiplied by scalars and so naturally form a vector space. Before we baptise our sequence space with a norm, let us be clear on which sequences are allowed in our space. Well, this is all a lie, as I have only talked about a singular sequence space, but in reality there are many different sequence spaces. Take for example the space of all sequences whose series is absolutely convergent, l_1, that is

    \[l_1=\{(a_n):\sum_{n=0}^{\infty}\abs{a_n}<\infty\},\]

and this has a norm very similar to the 1-norm

    \[\norm{(a_n)}_{l_1}=\sum_{n=0}^{\infty}\abs{a_n},\]

the difference being that since in our infinite sequence we have infinite terms, we sum from 0 to \infty, where as in \mathbb{R}^N, we only had to sum components up to N. Just like the p-norm, we can have any l_p space defined as

    \[l_p=\{(a_n):\sum_{n=0}^{\infty}\abs{a_n}^p<\infty\}\]

with the generalised p-norm

    \[\norm{(a_n)}_{l_p}=\big(\sum_{n=0}^{\infty}\abs{a_n}^p\big)^{\frac{1}{p}}.\]

We also define the set of all bounded sequences, l_{\infty} as

    \[l_{\infty}=\{(a_n): \sup_{1\leq i\leq\infty}\abs{a_i}<\infty\}\]

with the generalised infinite norm to be

    \[\norm{(a_n)}_{\infty}=\sup_{1\leq i\leq\infty}\abs{a_i}.\]

Note that these are all different spaces! They all have different sequences and convergence is different in each. Before we can look at that, we have to define what we mean by convergence in a normed vector space, and unfortunately, that will happen in the next post!

The final type of normed vector space I would like to consider are function spaces. We can add functions together and multiply them by scalars (point wise at least) and so they form a vector space. Just like the sequence spaces, we have to be clear on functions are in each space that we want. Consider for example the space of all continuous functions on the interval [0,1], C[0,1]. We christen this space with the norm

    \[\norm{f}=\sup_{x\in[0,1]}\abs{f(x)}.\]

There are also other types of function spaces such as L^p(A) where you are in the group (part of the cool kids) if

    \[\int_A\abs{f(x)}^pdx<\infty,\]

and if you are a cool kid, then your norm is given by

    \[\big(\int_A\abs{f(x)}^pdx\big)^{\frac{1}{p}}.\]

I could go on, but for now, this is sufficient for us to move on to the next stage of our wild ride. I hope you enjoyed the zoo of normed vector spaces, but as the cool kids said back in the day, “it’s not a party until Banach shows up.”  Here is a picture of Banach for reference.

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